15 September 2007

Why g ~ π2

The acceleration due to gravity, at the surface of the Earth, is about 9.81 m/s2. (If you are some of my students, you think it's 10, which is confusing for a moment. Fortunately none of my students thought it was 32.)

π2 = 9.87.

The approximate numerical equality of these numbers is not a coincidence.

I was reminded of this by Mark Dominus' post John Wilkins invents the meter, which I found via his post The Wilkins pendulum mystery resolved. In 1668, John Wilkins defined the meter to be essentially the length of the "seconds pendulum", i. e. the pendulum whose period is two seconds (the "mystery" there is that since one wants a pendulum with a light cord and a heavy bob, one has to take into account the moment of inertia of the heavy spherical bob to define this correctly). This turns out to be about 39.1 inches.

A meter is 39.37 inches.

Indeed, originally the length of the seconds pendulum was going to be the length of the meter, but the slightly different suggestion was adopted instead that the meter would be one part in 107 of the distance from the North Pole to the Equator via Paris. (Now, in fact, the meter is defined by assuming the second known and stating the speed of light.) Wikipedia tells me that these definitions were adopted in 1790, 1791, and 1983 respectively; there are others intermediate between the last two.)

The period of a pendulum is approximately T = 2π(L/g)1/2, where L is its length and g is the acceleration due to gravity. If we set T = 2 and L = 1, and solve for g, we get g = π2.

I suspect that the reason the seconds pendulum wasn't adopted is that the formula for the period involves the small-angle approximation; in reality the period of a pendulum depends not only on its length and on the acceleration due to gravity but also on the angle through which it swings. The period is given by

and so the fractional magnitude of the error is about 1/4 sin2 (θ/2); if θ is, say, one degree this is one part in about fifty thousand, or nearly two seconds a day. That error would become an error in the definition of the meter, and perhaps the Powers that Were thought that was a bit too much. (I'm not sure how accurate surveying was at the time, though, so I can't comment on the accuracy of their eventual definition based on the size of the Earth.)

That error would become an error in the definition of the meter, and perhaps the Powers that Were thought that was a bit too much. (I'm not sure how accurate surveying was at the time, though, so I can't comment on the accuracy of their eventual definition based on the size of the Earth.)

For a fascinating discussion of both these issues, I highly recommend The Measure of All Things, by Ken Alder. It's not always the most exciting book ever, but for anyone interested in the history of science, it's a must-read!

Yes, of course g is not exactly pi^2 m/s^2, because the length of a seconds pendulum was never actually adopted as the definition of the meter*! The definition that was adopted, however, was close enough that pi^2 m/s^2 is still a pretty good approximation.

Which part of Isabel's post aren't you getting?

* And because g varies depending on where on Earth you are... but let's not get into that. :)

it wouldn't surprise me to learn that they wanted to fix the length of the meter as the length of the seconds pendulum at Paris, mirroring the actual first definition of the meter as one ten-millionth of the distance from the North Pole to the equator through Paris.

Although g might not vary enough over the surface of the Earth for that to become an issue; the error in the small-angle approximation might swamp that.

In my humble opinion g ~ π² is the same coincidence as the fact that one ten-millionth of the length of the Earth's meridian along a quadrant ~ the length of a pendulum with a half-period of one second.

I'm sure you know, but you may want to point out that the small angle approximation isn't necessary at all - we can solve the equations of motions for the simple pendulum exactly _if_ we allow ourselves to use elliptic functions!

Because our functions live on the torus instead of the circle we get two periods instead of one - going through the math you can check that one of them is purely real (which corresponds to the actual, measured, period of the pendulum).

What is the other one? It is purely imaginary. If the pendulum is swinging out to a maximum angle of theta, the imaginary period tells you the measured period of a pendulum swinging out to pi minus theta!!

I guess this was one of the earliest applications of elliptic functions, and to me, is still one of the most fascinating!

I think folks are getting bogged down in what's a coincidence or not. It looks like Isabel is trying to tell us that g ~ π2 is because Wilkes essentially tried to define it that way in a roundabout way throught the definition of a mter, but a meter ended up to be defined using the Earth's surface instead and thus g is only close to π2.

arivero has the more interesting point - Why is g ~ π2 when the meter is defined the way it is? Using the Earth based length and time units (great idea!) you have the coincidence of ~1/40,000 showing up in both 1 meter and a 2 second pendulum.

The time depends on how fast the earth rotates, and that's related to the formation of the earth and whatever random smack made the moon and slowed the Earth, and not necessarily to any fundamental property of the universe.

What a cool post - I noticed g was very close to pi^2 when I was a kid, and always wondered why. So now I know. Why pi^2 is quite close to 10 was also something of an intriguing mystery, and can't be muddied by issues of units. (Noam Elkies has a nice little note about this.)Taking these together we have a splendid simplifying approximation for use with MKS units.

BTW, I'm not sure your finite amplitude analysis is quite right - maybe thetamax/2 should be replaced by sin^2(thetamax) - and the whole thing recognised as a complete elliptic integral, Old skool rules.